Global dynamics below the ground state energy for the Zakharov system in the 3D radial case (Q387896)

The authors study the global behavior of solutions for the Zakharov system NEWLINE\[NEWLINE\begin{cases} i\dot u-\bigtriangleup u=n\;u,\\ \ddot n/\alpha^2-\bigtriangleup n=-\bigtriangleup |u|^2,\end{cases}NEWLINE\]NEWLINE with the initial data \( u(0, x)= u_0, n(0, x) = n_0, \dot n(0, x) = n_1,\) in three space dimensions \((u, n)(t, x) : I \times\mathbb R^{1+3}\rightarrow : C \times\mathbb R,\) under the radial symmetry \((u, n) = (u, n)(t, |x|)\). This system describes the Langmuir turbulence in plasma, where \(u\) is the slow oscillation of the electric, \(n\) is the ion density fluctuation, and the constant \(\alpha > 0\) is the ion sound speed. For this system, the authors obtain dichotomy between the scattering and the grow up.